Problem 76 - Prealgebra

Question

Divide the decimal by the given power of 10 . \(\frac{520.77}{1000}\)

Step-by-Step Solution

Answer

The result of dividing 520.77 by 1000 is 0.52077.

Step 1: Identify Decimal Movement for Division by 1000

When dividing a number by 1000, move the decimal point to the left by three places, because 1000 is 10 to the power of 3.

Step 2: Apply the Decimal Movement

Start with the number 520.77. Move the decimal three places to the left. This will shift the decimal point from between 520 and 77, to the left of 5, resulting in 0.52077.

Key Concepts

Decimal Movement

Decimal movement is a fundamental concept when working with decimals and powers of 10. It helps you understand how numbers change when you multiply or divide by these powers.
When dividing by a power of 10, such as 10, 100, or 1000, you'll shift the decimal point to the left as many places as there are zeros in the power. In our example, we divide by 1000, which has three zeros. Thus, we shift the decimal point three places to the left.
If you have a number like 520.77, you simply visualize moving that dot three steps leftward:

• Step 1: Start at 520.77
• Step 2: Move one place to make it 52.077
• Step 3: Move another to make it 5.2077
• Step 4: Finally, move to 0.52077

It's an easy method once you get the hang of it. Remember, each power of 10 corresponds to how many times you move the decimal in that direction. Mastering this makes division by powers of 10 a breeze!

Powers of 10

Understanding powers of 10 is crucial for working with large and small numbers efficiently. A power of 10 represents 10 raised to an exponent, showing how many times you multiply 10 by itself. For example, 10 to the power of 3 (10⁴) means 1000.
In division, the role of powers of 10 is to guide us on how far to move the decimal. For instance, dividing by 1000 is also thought of as multiplying by 10 raised to the power of -3 (10⁻³). This negative exponent signifies division.
Recognizing these powers helps to quickly determine how numbers scale up or down. It's a consistent rule:

• 10⁰ = 1
• 10¹ = 10
• 10² = 100
• 10³ = 1000
• ...
• 10⁻¹ = 0.1 (Dividing by 10)
• 10⁻² = 0.01 (Dividing by 100)
• 10⁻³ = 0.001 (Dividing by 1000)

Keep these in mind as they repeat in arithmetic, making it fast and clear to adjust decimals, especially in division scenarios.

Place Value

Place value is all about understanding how the position of a digit in a number affects its value. This is particularly important in decimal numbers. Each position represents a different power of 10.
For instance, in the number 520.77:

• The 5 is in the hundreds place, representing 5 imes 100.
• The 2 is in the tens place, representing 2 imes 10.
• The 0 is in the ones place, representing 0 imes 1.
• The 7 right after the decimal represents 7 imes 0.1, showing it's in the tenths place.
• The final 7 represents 7 imes 0.01, indicating it's in the hundredths place.

Moving the decimal point alters the value associated with each digit because it shifts these place values. To divide by 1000, making the result 0.52077, the original place values drop, turning what was 520 into fractions of a whole. By comprehending place value's role, you can confidently read, understand, and manipulate decimal numbers across various mathematical tasks.